Yandex has been #1 in Russian speaking segment of the internet for the last 10 years or so. I use it almost every day. It looks like this expansion into English speaking market is just an experiment, and I'm not sure what they are trying to achieve. It doesn't look like they can compete with Google or Bing.

edit: btw, one of the interesting features Yandex provides and Google still doesn't is street view for Russian cities: http://maps.yandex.ru/-/CFfz7mA.